| L(s) = 1 | + (0.797 + 1.16i)2-s + (−0.727 + 1.86i)4-s + (2.42 − 0.426i)5-s + (−2.49 + 2.97i)7-s + (−2.75 + 0.636i)8-s + (2.42 + 2.48i)10-s + (0.0101 − 0.0573i)11-s + (4.33 + 1.57i)13-s + (−5.46 − 0.542i)14-s + (−2.94 − 2.71i)16-s + (−1.79 + 1.03i)17-s + (2.46 + 1.42i)19-s + (−0.965 + 4.82i)20-s + (0.0750 − 0.0339i)22-s + (4.37 − 3.67i)23-s + ⋯ |
| L(s) = 1 | + (0.564 + 0.825i)2-s + (−0.363 + 0.931i)4-s + (1.08 − 0.190i)5-s + (−0.943 + 1.12i)7-s + (−0.974 + 0.225i)8-s + (0.768 + 0.786i)10-s + (0.00305 − 0.0173i)11-s + (1.20 + 0.437i)13-s + (−1.46 − 0.144i)14-s + (−0.735 − 0.677i)16-s + (−0.434 + 0.250i)17-s + (0.564 + 0.326i)19-s + (−0.215 + 1.07i)20-s + (0.0160 − 0.00723i)22-s + (0.912 − 0.765i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08324 + 1.42458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08324 + 1.42458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.797 - 1.16i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.42 + 0.426i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.49 - 2.97i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0101 + 0.0573i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.33 - 1.57i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.79 - 1.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.46 - 1.42i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.37 + 3.67i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.37 + 3.76i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.90 + 5.84i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.41 - 4.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.375 + 1.03i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.50 - 1.67i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.18 + 3.51i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + (0.277 + 1.57i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.61 + 1.35i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.58 + 7.10i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.08 - 7.07i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.26 + 5.65i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.36 - 6.51i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 3.89i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.85 - 1.07i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.895 - 5.07i)T + (-91.1 - 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19663907855371637241260336655, −11.12024567930091777189721044051, −9.539585394496868465741904331098, −9.152733210118631079325687425717, −8.135420743258026152640033173141, −6.58998461627141425183952719154, −6.07453704942997269741124669740, −5.26788885082062633127077636699, −3.74239200559594783288781357739, −2.39530991614007499824642685285,
1.22543457315620430700952564120, 2.92869838562450571915127680372, 3.88102491893732474237355447856, 5.34434518395296591273435738479, 6.24067638646052741352850742713, 7.19894105299374367785541627960, 9.072827395458744864867604632417, 9.614603937207707938150594012376, 10.70719276981352819835738510279, 10.96898744922682485203246347127
